Theorem neisspw 12788

Description: The neighborhoods of any set are subsets of the base set. (Contributed by Stefan O'Rear, 6-Aug-2015.)

Hypothesis

Ref	Expression
neifval.1	|- X = U. J

Assertion

Ref	Expression
neisspw	|- ( J e. Top -> ( ( nei ` J ) ` S ) C_ ~P X )

Proof of Theorem neisspw

Dummy variable v is distinct from all other variables.

Step	Hyp	Ref	Expression
1		neifval.1	. . . . 5 |- X = U. J
2	1	neii1 12787	. . . 4 |- ( ( J e. Top /\ v e. ( ( nei ` J ) ` S ) ) -> v C_ X )
3		velpw 3566	. . . 4 |- ( v e. ~P X <-> v C_ X )
4	2, 3	sylibr 133	. . 3 |- ( ( J e. Top /\ v e. ( ( nei ` J ) ` S ) ) -> v e. ~P X )
5	4	ex 114	. 2 |- ( J e. Top -> ( v e. ( ( nei ` J ) ` S ) -> v e. ~P X ) )
6	5	ssrdv 3148	1 |- ( J e. Top -> ( ( nei ` J ) ` S ) C_ ~P X )

Syntax hints:   -> wi 4   /\ wa 103   = wceq 1343   e. wcel 2136   C_ wss 3116   ~Pcpw 3559   U.cuni 3789   ` cfv 5188   Topctop 12635   neicnei 12778

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 699  ax-5 1435  ax-7 1436  ax-gen 1437  ax-ie1 1481  ax-ie2 1482  ax-8 1492  ax-10 1493  ax-11 1494  ax-i12 1495  ax-bndl 1497  ax-4 1498  ax-17 1514  ax-i9 1518  ax-ial 1522  ax-i5r 1523  ax-14 2139  ax-ext 2147  ax-coll 4097  ax-sep 4100  ax-pow 4153  ax-pr 4187

This theorem depends on definitions:  df-bi 116  df-3an 970  df-tru 1346  df-nf 1449  df-sb 1751  df-eu 2017  df-mo 2018  df-clab 2152  df-cleq 2158  df-clel 2161  df-nfc 2297  df-ral 2449  df-rex 2450  df-reu 2451  df-rab 2453  df-v 2728  df-sbc 2952  df-csb 3046  df-un 3120  df-in 3122  df-ss 3129  df-pw 3561  df-sn 3582  df-pr 3583  df-op 3585  df-uni 3790  df-iun 3868  df-br 3983  df-opab 4044  df-mpt 4045  df-id 4271  df-xp 4610  df-rel 4611  df-cnv 4612  df-co 4613  df-dm 4614  df-rn 4615  df-res 4616  df-ima 4617  df-iota 5153  df-fun 5190  df-fn 5191  df-f 5192  df-f1 5193  df-fo 5194  df-f1o 5195  df-fv 5196  df-top 12636  df-nei 12779

This theorem is referenced by: (None)